Petrie conjecture

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Introduction

The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$${{Stub}}== Introduction == <wikitex>; The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly and non-trivially on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular. Petrie restricted his attention to smooth actions of the Lie group $S^1$, the circle {{cite|Petrie1972}} (or more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}}) on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $M$ is a closed smooth n$-manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts smoothly and non-trivially on $M$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ {{endthm}} Petrie {{cite|Petrie1973}} has shown that the total Pontrjagin class of $M$ agrees with that of $\CP^n$ if the action of $S^1$ on $M$ extends to a smooth action of $T^n$. The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}. According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with the first Chern class $c_1(M) = (n+1)x$. Another special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and {{cite|Masuda1981}}. Moreover, by the work of {{cite|Dessai2002}}, the Petrie conjecture holds if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$. Similarly, the work of {{cite|Dessai&Wilking2004}} confirms the Petrie conjecture under the assumption that $\dim M \leq 8k-4$ and the action of $S^1$ on $M$ extends to a smooth action of $T^k$ for $k \geq 1$. A symplectic version of the Petrie conjecture is discussed in the article {{cite|Tolman2010}}. == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]] </wikitex>G$ acts smoothly and non-trivially on a closed smooth manifold

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$M$, what constraints does this place on the topology of

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$M$ in general and on the Pontrjagin classes of

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$M$ in particular.

Petrie restricted his attention to smooth actions of the Lie group $S^1$$S^1$, the circle [Petrie1972] (or more generally, the torus $T^k$$T^k$ for $k \geq 1$$k \geq 1$ [Petrie1973])

on closed smooth manifolds

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$M$ which are homotopy equivalent to $\CP^n$$\CP^n$.

Conjecture 0.1 [Petrie1972].

Suppose that

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$M$ is a closed smooth $2n$$2n$-manifold homotopy equivalent to $\CP^n$$\CP^n$ and that $S^1$$S^1$ acts smoothly and non-trivially on

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$M$. Then the total Pontrjagin class $p(M)$$p(M)$ of

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$M$ agrees with that of $\CP^n$$\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$$x \in H^2(M; \mathbb{Z})$,

$\displaystyle p(M) = (1+x^2)^{n+1}.$ $$\displaystyle p(M) = (1+x^2)^{n+1}.$$

Petrie [Petrie1973] has shown that the total Pontrjagin class of

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$M$ agrees with that of $\CP^n$$\CP^n$ if the action of $S^1$$S^1$ on

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$M$ extends to a smooth action of $T^n$$T^n$.

The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$$\dim M \leq 8$, the statement is true by the work of [Dejter1976] and [James1985].

According to [Hattori1978], the conjecture is also true if

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$M$ admits an invarint almost complex structure with the first Chern class $c_1(M) = (n+1)x$$c_1(M) = (n+1)x$. Another special cases

where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and

[Masuda1981]. Moreover, by the work of [Dessai2002], the Petrie conjecture holds if $\dim M \leq 22$$\dim M \leq 22$ and the action of $S^1$$S^1$ on

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$M$ extends to an appropriate action of $Pin(2)$$Pin(2)$. Similarly, the work of [Dessai&Wilking2004] confirms the Petrie conjecture under the assumption that $\dim M \leq 8k-4$$\dim M \leq 8k-4$ and the action of $S^1$$S^1$ on

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$M$ extends to a smooth action of $T^k$$T^k$

for $k \geq 1$$k \geq 1$.

A symplectic version of the Petrie conjecture is discussed in the article [Tolman2010].

References

• [Dejter1976] I. J. Dejter, Smooth $S^1$$S^1$-manifolds in the homotopy type of $CP^3$$CP^3$, Michigan Math. J. 23 (1976), no.1, 83–95. MR0402789 (53 #6603) Zbl 0326.57009
• [Dessai&Wilking2004] A. Dessai and B. Wilking, Torus actions on homotopy complex projective spaces, Math. Z. 247 (2004), no.3, 505–511. MR2114425 (2006c:57033) Zbl 1068.57034
• [Dessai2002] A. Dessai, Homotopy complex projective spaces with $Pin(2)$$Pin(2)$-action, Topology Appl. 122 (2002), no.3, 487–499. MR1911696 (2003f:58048) Zbl 0998.57048
• [Hattori1978] A. Hattori, $Spin^c$$Spin^c$-structures and $S^1$$S^1$-actions, Invent. Math. 48 (1978), no.1, 7–31. MR508087 (80e:57051) Zbl 0395.57020
• [Iberkleid1978] W. Iberkleid, Pseudolinear spheres, Michigan Math. J. 25 (1978), no.3, 359–370. MR512906 (80d:57023) Zbl 0377.57008
• [James1985] D. M. James, Smooth $S^1$$S^1$ actions on homotopy $\mathbb{C}P^4$$\mathbb{C}P^4$'s, Michigan Math. J. 32 (1985), no.3, 259–266. MR803831 (87c:57031) Zbl 0602.57026
• [Masuda1981] M. Masuda, On smooth $S^1$$S^1$-actions on cohomology complex projective spaces. The case where the fixed point set consists of four connected components, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no.1, 127–167. MR617869 (82i:57031) Zbl 0462.57019
• [Muslin1978] Template:Muslin1978
• [Petrie1972] T. Petrie, Smooth $S^1$$S^1$-actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR0296970 Zbl 0247.57010
• [Petrie1973] T. Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973), 139–146. MR0322893 (48 #1254) Zbl 0262.57021
• [Tolman2010] S. Tolman, On a symplectic generalization of Petrie's conjecture, Trans. Amer. Math. Soc. 362 (2010), no.8, 3963–3996. MR2638879 Zbl 1216.53074
• [Tsukada&Washiyama1979] E. Tsukada and R. Washiyama, $S^1$$S^1$-actions on cohomology complex projective spaces with three components of the fixed point sets, Hiroshima Math. J. 9 (1979), no.1, 41–46. MR529325 (80j:57043) Zbl 0411.57037
• [Wang1975] K. Wang, Differentiable circle group actions on homotopy complex projective spaces, Math. Ann. 214 (1975), 73–80. MR0372895 (51 #9099) Zbl 0285.57025
• [Yoshida1976] Template:Yoshida1976